Showing papers on "Hydrostatic equilibrium published in 2004"

An asymptotic theory for the interaction of waves and currents in coastal waters

Abstract: A multi-scale asymptotic theory is derived for the evolution and interaction of currents and surface gravity waves in water of finite depth, under conditions typical of coastal shelf waters outside the surf zone. The theory provides a practical and useful model with which wave–current coupling may be explored without the necessity of resolving features of the flow on space and time scales of the primary gravity-wave oscillations. The essential nature of the dynamical interaction is currents modulating the slowly evolving phase of the wave field and waves providing both phase-averaged forcing of long infra-gravity waves and wave-averaged vortex and Bernoulli-head forces and hydrostatic static set-up for the low-frequency current and sea-level evolution equations. Analogous relations are derived for wave-averaged material tracers and density stratification that include advection by horizontal Stokes drift and by a vertical Stokes pseudo-velocity that is the incompressible companion to the horizontal Stokes velocity. Illustrative solutions are analysed for the special case of depth-independent currents and tracers associated with an incident surface wave field and a vortex with O(1) Rossby number above continental shelf topography.

A dynamical model for the distribution of dark matter and gas in galaxy clusters

Abstract: Using the results of an extended set of high-resolution non-radiative hydrodynamic simulations of galaxy clusters, we obtain simple analytic formulae for the dark matter and hot gas distribution, in the spherical approximation. Starting from the dark matter phase-space radial density distribution, we derive fits for the dark matter density, velocity dispersion and velocity anisotropy. We use these models to test the dynamical equilibrium hypothesis through the Jeans equation: we find that this is satisfied to good accuracy by our simulated clusters inside their virial radii. This result also shows that our fits constitute a self-consistent dynamical model for these systems. We then extend our analysis to the hot gas component, obtaining analytic fits for the gas density, temperature and velocity structure, with no further hypothesis on the gas dynamical status or state equation. Gas and dark matter show similar density profiles down to ≈0.06Rv (with Rv the virial radius), while at smaller radii the gas flattens, producing a central core. Gas temperatures are almost isothermal out to roughly 0.2 Rv, then steeply decrease, reaching at the virial radius a value almost a factor of 2 lower. We find that the gas is not at rest inside Rv: velocity dispersions are increasing functions of the radius, motions are isotropic to slightly tangential, and contribute non-negligibly to the total pressure support. We test this model using a generalization of the hydrostatic equilibrium equation, where the gas motion is properly taken into account. Again we find that the fits provide an accurate description of the system: the hot gas is in equilibrium and is a good tracer of the overall cluster potential if all terms (density, temperature and velocity) are taken into account, while simpler assumptions cause systematic mass underestimates. In particular, we find that using the so-called β-model underestimates the true cluster mass by up to 50 per cent at large radii. We also find that, if gas velocities are neglected, then a simple isothermal model fares better at large radii than a non-isothermal one. The shape of the gas density profile at small radii is at least partially explained by the gas expansion caused by energy transfer from dark matter during the collapse. In fact, when gas bulk energy is also considered, gas and dark matter are in energy equipartition in the final system at radii r > 0.1Rv, while at smaller radii the gas is hotter than the dark matter. This energy imbalance is also probably the reason of the further global halo compression compared with a pure collisionless collapse, which we point out by comparing the dark matter and total density profiles of our hydro-simulated clusters with a set of identical – but pure N-body – ones. The compression has the effect of raising the mean concentration by an amount of roughly 10 per cent.

Generalized Collapse Solutions with Nonzero Initial Velocities for Star Formation in Molecular Cloud Cores

Abstract: Motivated by recent observations that show that starless molecular cloud cores exhibit subsonic inward velocities, we revisit the collapse problem for polytropic gaseous spheres. In particular, we provide a generalized treatment of protostellar collapse in the spherical limit and find semianalytic (self-similar) solutions, corresponding numerical solutions, and purely analytic calculations of the mass infall rates (the three approaches are in good agreement). This study focuses on collapse solutions that exhibit nonzero inward velocities at large radii, as observed in molecular cloud cores, and extends previous work in four ways: (1) the initial conditions allow nonzero initial inward velocity, (2) the starting states can exceed the density of hydrostatic equilibrium so that the collapse itself can provide the observed inward motions, (3) we consider different equations of state, especially those that are softer than isothermal, and (4) we consider dynamic equations of state that are different from the effective equation of state that produces the initial density distribution. This work determines the infall rates over a wide range of parameter space, as characterized by four variables: the initial inward velocity v∞, the overdensity Λ of the initial state, the index Γ of the static equation of state, and the index γ of the dynamic equation of state. For the range of parameter space applicable to observed cores, the resulting infall rate is about a factor of 2 larger than that found in previous theoretical studies (those with hydrostatic initial conditions and v∞ = 0).

Atmosphere–Ocean Modeling Exploiting Fluid Isomorphisms

Abstract: Mathematical isomorphisms between the hydrostatic equations that govern the evolution of a compressible atmosphere and an incompressible ocean are described and exploited to guide the design of a hydrodynamical kernel for simulation of either fluid.

Triaxial Test Simulations with Discrete Element Method and Hydrostatic Boundaries

Abstract: A new boundary condition has been developed for the discrete element method. This boundary is different from the conventional periodic, rigid, or flexible boundries. This new boundary mechanism was developed to simulate triaxial tests. The new boundary, hydrostatic boundary, simulated the chamber fluid but not the rubber membrane. When a particle (ellipsoids in our simulations) contacts the hydrostatic boundary, pressure is developed. The interaction between the particle and the boundary is calculated analytically based on geometry. This hydrostatic boundary condition was implemented into an existing ellipsoidal discrete element code. Triaxial compression drained tests were performed with both periodic and hydrostatic boundaries. The result showed an increase in friction angle over the values observed from the periodic boundary mechanism. The result also closely resembles the experimental triaxial data. Thirteen specimens were generated and were used to investigate the following variables: particle shape, specimen size, and void ratio. A unique slope of the linear relationship between friction angle and void ratio was identified for monosize specimens of different particle shapes. It is found that the friction angle decreases as the aspect ratio increases provided that the void ratio of the two specimens is the same. The friction angle is linear proportional to the coordination number for monosize specimens regardless the specimen size. Also, the specimen size does not influence the behavior of two-size specimens.

Hydrostatic and non-hydrostatic studies of gravitational adjustment over a slope

Abstract: In many numerical ocean models, the hydrostatic approximation is made. This approximation causes a considerable saving in computing time. However, for phenomena involving large vertical speeds, for many small scale phenomena, and in areas with weak stratification, the approximation becomes questionable. In this report, a σ-coordinate hydrostatic C-grid model is extended to include non-hydrostatic dynamics. The test cases involve gravitational adjustment of a downslope flow. The first test case has a simplified slope profile and no ambient stratification in the deep basin. The second test case has ambient stratification and more realistic topography. The differences between hydrostatic and non-hydrostatic simulations are described and discussed. It is shown that the shapes of the head and the body of density driven plumes are better preserved in the non-hydrostatic experiments. The wave propagation away from the plume head is considerably reduced when including non-hydrostatic effects.

How Sensitive are Coarse General Circulation Models to Fundamental Approximations in the Equations of Motion

Abstract: The advent of high-precision gravity missions presents the opportunity to accurately measure variations in the distribution of mass in the ocean. Such a data source will prove valuable in state estimation and constraining general circulation models (GCMs) in general. However, conventional GCMs make the Boussinesq approximations, a consequence of which is that mass is not conserved. By use of the height‐pressure coordinate isomorphism implemented in the Massachusetts Institute of Technology general circulation model (MITGCM), the impact of non-Boussinesq effects can be evaluated. Although implementing a non-Boussinesq model in pressure coordinates is relatively straightforward, making a direct comparison between height and pressure coordinate (i.e., Boussinesq and non-Boussinesq) models is not simple. However, a careful comparison of the height coordinate and the pressure coordinate solutions ensures that only non-Boussinesq effects can be responsible for the observed differences. As a yardstick, these differences are also compared with those between the Boussinesq hydrostatic and models in which the hydrostatic approximation has been relaxed, another approximation commonly made in GCMs. Model errors (differences) caused by the Boussinesq and hydrostatic approximations are demonstrated to be of comparable magnitude. Differences induced by small changes in subgrid-scale parameterizations are at least as large. Therefore, non-Boussinesq and nonhydrostatic effects are most likely negligible with respect to other model uncertainties. However, because there is no additional cost incurred in using a pressure coordinate model, it is argued that non-Boussinesq modeling is preferable simply for tidiness. It is also concluded that even coarse-resolution GCMs can be sensitive to small perturbations in the dynamical equations.

Structure-Dependent Hydrostatic Deformation Potentials of Individual Single-Walled Carbon Nanotubes

Abstract: Summary: The hydrostatic pressure coefficients of interband transition energies of a number of single-walled carbon nanotubes with different chiralities were measured. Both optical absorption and photoluminescence experiments were performed on de-bundled, single-walled carbon nanotube suspensions with hydrostatic pressure applied by diamond anvil cells. The pressure coefficients of the first van Hove transition (bandgap) energies are negative and dependent on the nanotube structure, while the second van Hove transitions are much less sensitive to hydrostatic pressure. The hydrostatic deformation potentials of individual nanotubes are deduced within an elastic model. An empirical equation that relates the pressure coefficients to nanotube structure is presented and discussed.

Derivation of the Equations of Atmospheric Motion in Oblate Spheroidal Coordinates

Abstract: Since Earth is more nearly an oblate spheroid than a sphere, it is of at least theoretical interest to develop the atmospheric equations of motion in spheroidal coordinates. In this system the horizontal unit vectors are oriented eastward and northward along the surfaces of ellipsoids, while the orthogonal unit vector is oriented vertically along the surfaces of intersecting confocal hyperboloids. Using the theory of orthogonal curvilinear coordinates, the spheroidal equations of relative atmospheric motion are derived from the vector equation of absolute motion. With the exception of two terms in the meridional and vertical equations of motion that are unique to the spheroidal system, all of the metric and rotational terms in the spheroidal system correspond to those found in the familiar spherical formulation, but now have coefficients that are functions of both the spheroidal latitude and elevation. The unique spheroidal terms arise from the resolution of the difference between the directions of apparent gravity and Newtonian gravitation, which is neglected in the spherical formulation. The complete spheroidal equations conserve both absolute angular momentum and total kinetic energy, and in the limit as Earth’s focal distance or eccentricity approaches zero, reduce to the familiar spherical equations in both the general and hydrostatic cases. The differences between solutions of the spheroidal and spherical equations are not expected to be significant in most applications, although there is the possibility of the accumulation of systematic differences in long-term integrations.

Thin-shell magnetohydrodynamic equations for the solar tachocline

Abstract: We present a new system of equations designed to study global-scale dynamics in the stably-stratified portion of the solar tachocline. This system is derived from the 3D equations of magnetohydrodynamics in a rotating spherical shell under the assumption that the shell is thin and stably-stratified (subadiabatic). The resulting thin-shell model can be regarded as a magnetic generalization of the hydrostatic primitive equations often used in meteorology. It is simpler in form than the more general anelastic or Boussinesq equations, making it more amenable to analysis and interpretation and more computationally efficient. However, the thin-shell system is still three-dimensional and as such represents an important extension to previous 2D and shallow-water approaches. In this paper we derive the governing equations for our thin-shell model and discuss its underlying assumptions, its context relative to other models, and its application to the solar tachocline. We also demonstrate that the dissipationless thin-shell system conserves energy, angular momentum and magnetic helicity.

The Effect of Wind Shear and Curvature on the Gravity Wave Drag Produced by a Ridge

Abstract: The analytical model proposed by Teixeira, Miranda, and Valente is modified to calculate the gravity wave drag exerted by a stratified flow over a 2D mountain ridge. The drag is found to be more strongly affected by the vertical variation of the background velocity than for an axisymmetric mountain. In the hydrostatic approximation, the corrections to the drag due to this effect do not depend on the detailed shape of the ridge as long as this is exactly 2D. Besides the drag, all the perturbed quantities of the flow at the surface, including the pressure, may be calculated analytically.

Hydrostatic and non-hydrostatic internal wave models

Abstract: Numerical models have become an indispensable tool for ocean and inland flow modeling. Such models typically use the hydrostatic approximation based on the argument that their horizontal length scales are longer than the vertical length scales. There are a wide variety of physical processes in oceans and inland water systems, and many of these processes are adequately modeled with the hydrostatic approximation. However, internal waves contribute to the physics that influence mixing in a density stratified system and have been previously shown to be nonhydrostatic. The neglect of non-hydrostatic pressure in a hydrostatic model is problematic since non-hydrostatic pressure plays a significant role in internal wave evolution balancing nonlinear wave steepening. Where non-hydrostatic pressure is neglected in a model, the governing equations are missing a piece of the physics that control the internal wave evolution, so it should not be surprising that the evolution may be poorly predicted. Despite the knowledge that the non-hydrostatic pressure is necessary for correctly modeling the physics of a steepening internal wave, the high computational cost of solving the non-hydrostatic pressure has limited its use in large-scale systems. Furthermore, the errors associated with hydrostatic modeling of internal waves have not been quantified. This research quantifies the differences between hydrostatic and non-hydrostatic simulations of internal wave evolution and develops a method to a priori determine regions with non-hydrostatic behavior. In quantifying the errors and differences between the two models this research provides the characteristics of model error with grid refinement. Additionally, it is shown that hydrostatic models may develop high wavenumber “soliton-like” features that are purely a construct of model error, but may seem to mimic physical behaviors of the non-hydrostatic system. Finally, it is shown that regions of significant non-hydrostatic pressure gradients can be identified from a hydrostatic model. This latter finding is a building block towards coupling local nonhydrostatic solutions with global hydrostatic solutions for more efficient computational methods. The work presented here provides the foundations for future non-hydrostatic model development and application.

Three-Dimensional Structure of Forced Gravity Waves and Lee Waves

Abstract: The three-dimensional structure of lee waves is investigated using a combination of linear analysis and numerical simulation. The forcings are represented by flow over a single wave (monochromatic) in the along-stream direction but of limited extent in the cross-stream direction, and by flow over isolated obstacles. The flow structures considered are of constant static stability, and zero, positive, and negative basic-flow shears. Both nonhydrostatic and hydrostatic regimes are studied. Particular emphasis is placed on 1) the cross-stream structure of the waves, 2) the transition from three-dimensional to two-dimensional flow as the breadth of the obstacle is increased, 3) the criteria for three-dimensional nonhydrostatic to hydrostatic transitions, and 4) the effect of obstacle breadth-to-length aspect ratio on the wave drag for this linear system. It is shown that these aspects can in part be understood by relating the gravity waves produced by narrow-breadth obstacles to the “St. Andrew's Cros...

Structuring and support by Alfvén waves around prestellar cores

Abstract: Observations of molecular clouds show the existence of starless, dense cores, threaded by magnetic fields. Observed line widths indicate these dense condensates to be embedded in a supersonically turbulent environment. Under these conditions, the generation of magnetic waves is inevitable. In this paper, we study the structure and support of a 1D plane-parallel, self-gravitating slab, as a monochromatic, circularly polarized Alfven wave is injected in its central plane. Dimensional analysis shows that the solution must depend on three dimensionless parameters. To study the nonlinear, turbulent evolution of such a slab, we use 1D high resolution numerical simulations. For a parameter range inspired by molecular cloud observations, we find the following. 1) A single source of energy injection is sufficient to force persistent supersonic turbulence over several hydrostatic scale heights. 2) The time averaged spatial extension of the slab is comparable to the extension of the stationary, analytical WKB solution. Deviations, as well as the density substructure of the slab, depend on the wave-length of the injected wave. 3) Energy losses are dominated by loss of Poynting-flux and increase with increasing plasma beta. 4) Good spatial resolution is mandatory, making similar simulations in 3D currently prohibitively expensive.

Experimental study of void space, permeability and elastic anisotropy in crustal rocks under ambient and hydrostatic pressure.

Abstract: Anisotropy in the physical and transport properties of crustal rocks is a key influence on crustal evolution and energy resource management. Data from deep seismic soundings, borehole logging and laboratory measurement all show that the physical properties of the earth are anisotropic. Such anisotropy generally results from the superposition of fabric development during diagenesis and/or petrogenesis, and the application of anisotropic tectonic stresses. This leads to an aligned crack and pore fabric in crustal rocks that, in turn, leads to seismic velocity anisotropy and permeability anisotropy. This thesis describes an experimental study which aims to investigate the relationships between pressure, pore fabric geometry and seismic and permeability anisotropy under hydrostatic pressures from room pressure to ~4km depth equivalence within the Earth's crust. Firstly, pore fabric analyses of three representative crustal rock types is presented. These rock types represent a range of crack and pore fabrics. The average void space shape and orientation is determined 3-D using the methods of anisotropy of magnetic susceptibility and velocity anisotropy. Scanning electron microscopy and fluorescent-dye crack imaging techniques further aid in the void space characterisation. Secondly, the development and application of an apparatus capable of contemporaneously measuring elastic wave velocity, porosity and permeability at effective pressures of up to 100 MPa is described. Results are analysed in terms of applied effective pressure and the rock pore fabric type and orientation. Finally, the laboratory data are used to test models that attempt to predict geophysical parameters such as permeability and elastic wave velocity from microstructural attributes. This multi-facetted analysis allows a number of conclusions to be drawn, expanding the state-of-the-art in how the pore fabric microstructure of crustal rock is represented by the methods of elastic wave velocity and porosity, with reference to the hydrostatic confining pressure and hence the burial conditions of the rock.

Hydrostatic equilibrium of insular, static, spherically symmetric, perfect fluid solutions in general relativity

Abstract: An analysis of insular solutions of Einstein's field equations for static, spherically symmetric, source mass, on the basis of exterior Schwarzschild solution is presented. Following the analysis, we demonstrate that the regular solutions governed by a self-bound (that is, the surface density does not vanish together with pressure) equation of state (EOS) or density variation cannot exist in the state of hydrostatic equilibrium, because the source mass which belongs to them, does not represent the "actual mass" appears in the exterior Schwarzschild solution. The only configuration which could exist in this regard is governed by the homogeneous density distribution (i.e. the interior Schwarzschild solution). Other structures which naturally fulfill the requirement of the source mass, set up by exterior Schwarzschild solution (and, therefore, can exist in hydrostatic equilibrium) are either governed by gravitationally-bound regular solutions (i.e. the surface density also vanishes together with pressure), or self-bound singular solutions (i.e. the pressure and density both become infinity at the centre).

The dynamic method of determination of the piezoelectric hydrostatic coefficients

Abstract: The dynamic method of the determination of the piezoelectric hydrostatic coefficients, d h , was improved. We have constructed new equipment for more exact measurement by means of the direct dynamic method. The piston for the high-pressure mechanical excitation of pressure changes was used. The advantage of the proposed method is a low frequency (about 1 Hz) pressure excitation. The lever hydraulic press is able to create the hydrostatic pressure inside the pressure chamber up to 70 MPa. The temperature control is realised by the PID temperature controller with a resistivity heater and by the compressor cooler.

The hydrostatic equilibrium and Tsallis equilibrium for self-gravitating systems

Abstract: Self-gravitating systems are generally thought to behavior non-extensively due to the long-range nature of gravitational forces. We obtain a relation between the nonextensive parameter q of Tsallis statistics, the temperature gradient and the gravitational potential based on the equation of hydrostatic equilibrium of self-gravitating systems. It is suggested that the nonextensive parameter in Tsallis statistics has a clear physical meaning with regard to the non-isothermal nature of the systems with long-range interactions and Tsallis equilibrium distribution for the self-gravitating systems describes the property of hydrostatic equilibrium of the systems.

Quasi-hydrostatic intracluster gas under radiative cooling

Abstract: Quasi-hydrostatic cooling of the intracluster gas is studied. In the quasi-hydrostatic model, work done by gravity on the inflow gas with dP ¬= 0, where P is the gas pressure, is taken into account in the thermal balance. The gas flows in from the outer part so as to compensate the pressure loss of the gas undergoing radiative cooling, but the mass flow is so moderate and smooth that the gas is considered to be quasi-hydrostatic. The temperature of the cooling gas decreases toward the cluster center, but, unlike cooling flows with dP = 0, approaches a constant temperature of ∼1/3 the temperature of the non-cooling ambient gas. This does not mean that gravitational work cancels out radiative cooling, but means that the temperature of the cooling gas appears to approach a constant value toward the cluster center if the gas maintains the quasi-hydrostatic balance. We discuss the mass flow in quasi-hydrostatic cooling, and compare it with the standard isobaric cooling flow model. We also discuss the implication of M for the standard cooling flow model.

An isostatically compensated gravity model using spherical layer distributions

Abstract: A new isostatic model for the Earth's gravity field is presented based on a simple hypothesis of layers approximating constant density contrasts. The spherical layer distribution used to describe the hydrostatic equilibrium of the Earth's masses leads to a new set of spherical harmonic coefficients for the gravitational potential. First attempts to quantify the information content of these coefficients led to the outcome that they seem to explain the observed gravity field for a certain wavelength band, while they are insufficient for short and very long wavelengths. A synthesis of the derived coefficients over specific degree ranges provided a computation of band-limited geoid undulations on a global scale. The association of these potential quantities with known tectonic structures, such as the topography of the core-mantle boundary, strengthens the belief that the interpretation of Earth gravity models, especially those arising from global digital elevation models, should be considered in close relation with deep-Earth structure.

Representative Air Temperature of Thermally Heterogeneous Urban Areas Using the Measured Pressure Gradient

Abstract: A method to measure an area-averaged ground air temperature based on the hydrostatic equation is shown. The method was devised to overcome the problem of finding the most representative surface air temperature over a wide region, a problem that has seriously hindered the description of urban heat islands. The vertical pressure gradient is used and the hydrostatic equation is applied to estimate the average air temperature between two barometers, which is here called the hydrostatic temperature. The error analysis shows that the hydrostatic temperature can be estimated with a systematic error of 1.88C and a random error of 0.78C in the case in which the two barometers have a vertical separation of 228 m. The measured hydrostatic temperature agreed with the average of the directly measured temperature within 0.78C rms. For this barometer separation, the representative area of the hydrostatic temperature was experimentally found to be a 12-km-radius circle. The size of this area decreased when the vertical separation of the barometers decreased. The hydrostatic temperature is compared with the average directly measured temperature for various areas. The maximum correlation between them occurred for a circular area with a 12-km radius centered on the pressure measurements. The size of the representative area for this method is larger than that for the direct measurement of air temperature.

3D-simulation of the Outer Convection-zone of an A-star

Abstract: The convection code of Nordlund & Stein has been used to evaluate the 3D, radiation- coupled convection in a stellar atmosphere with Teff = 7300 K, logg = 4.3 and (Fe/H)= 0.0, corresponding to a main-sequence A9-star. I will present preliminary comparisons between the 3D-simulation and a conventional 1D stellar structure calculation, and elaborate on the conse- quences of the differences. From 3 dimensional simulations of convection, it has been known for the last two decades that convection grows stronger with increasing effective temperature and de- creasing gravity. By stronger, is here meant larger Mach-numbers, larger turbulent- to total-pressure ratios and larger convective fluctuations in temperature and density. The stronger convection has also been accompanied by increasing departures from 1D stellar models that fail to predict the extensive overshoot into the high atmosphere, the turbu- lent pressure and its effect on the hydrostatic equilibrium, the temperature fluctuations and the coupling with the highly non-linear opacity. The latter has the effect of heating the layers below the photosphere, thereby expanding the atmosphere, as also done by the turbulent pressure. The various 1D convection theories/formulations, e.g., classical mixing-length (Bohm-Vitense 1958), non-local extensions to it (Gough 1977) or an in- dependent formulation based on turbulence (Canuto & Mazzitelli 1992), all have similar shortcomings with respect to the simulations. Their predictive power is further limited by the free parameters involved. Going towards earlier type stars, apart from stronger convection, also means a more shallow outer convection zone. This combination is rather unpredictable and is the main motivation for the work presented here. Classical predictions call for the outer convection zone to disappear close to the transition between A and F stars, but details about where and how this transition occurs can only be gained from realistic, 3D simulations, as outlined below.

Some inferences on turbulence generation by gravity waves

Abstract: [1] We analyze data from 17 high-resolution thermosonde observations during a campaign near Adelaide, Australia, in 1998. We are able to explain the observed turbulence fractions by extending the turbulence model of Hines [1991], which explains turbulence by instabilities due to freely propagating independent gravity waves, from the hydrostatic wave regime toward lower frequencies. In a Monte Carlo simulation we show that the dependency of the observed gravity wave spectrum on vertical wave number is also consistent with the distribution of turbulent layer thickness as observed by the high-resolution thermosondes.

Pressure distribution in a curvilinear hydrostatic bearing lubricated by a micropolar fluid in the presence of a cross magnetic field

Abstract: The flow of a micropolar fluid in the clearance of a curvilinear hydrostatic thrust bearing in the presence of a cross magnetic field was considered. By solving the equations of motion for the orthogonal curvilinear coordinate system, a formula describing the pressure distribution was obtained. Plane and spherical thrust bearings were considered as examples.

Corrections to Solar Thermal Structure when a Turbulent Magnetic Field is Included

Abstract: Correction of non-ideal effect due to a magnetic fluctuating tensor is derived from the ideal MHD equations. The inclusion of a magnetic turbulent field leads to modifications of the hydrostatic equilibrium equation and thermodynamical variables such as the temperature T, the adiabatic exponent gamma, the adiabatic temperature gradient del(ad) and the temperature gradient del. In particular, the modifications in the adiabatic and radiative temperature gradients will result in a change in the Schwarzchild criterion, hence in the location of the base of the convective zone. Incorporating the modifications, we construct a modified thermodynamical equilibrium structure of the Sun.

Gravitational differentiation of liquid cores of planets and natural satellites

Abstract: Initial equations are obtained, similarity criteria are estimated and a project of simulation experiment is proposed for the gravitational dierentiation of liquid cores of planets and natural satellites. It is assumed that, first, the liquid core in an adiabatic state without thermal convection and, second, the inner solid core grows during the crystallization of a heavy component from the liquid core in such a way that the buoyancy force acting on a lighter component is directed strictly along the radius. It is also assumed that the radial distribution of density in the liquid core does not change during the time interval considered. These three natural assumptions enable an analytical description of basic hydrostatic eects controlling slow growth of the solid core, gravitational stratification of the liquid core, and sources of related compositional convection. The similarity criteria of such convection are mostly the same as for thermal convection. Additional criteria are the concentration contrast ( 1/10 in the Earth), the compressibility of the liquid core ( 10%), and the thickness of a concentration boundary layer ( 10 7 ) that, controlling the freezing-out of the liquid at the inner sphere, can give rise to asymmetry of the solid core. The excitation threshold of the compositional convection is much higher than a similar threshold for thermal convection, and the compositional convection itself can arise only at an intermediate stage of the gravitational dierentiation of the core. Observed magnetic fields are largely due to compositional convection in the Earth's core and, probably, in deep interiors of Mercury. At the contemporary evolutionary stage of Venus' interiors, the intensity of compositional convection is most likely insucient for the magnetic field excitation and it is undoubtedly too weak in the Mars' interiors.

Regularization by monotone perturbations of the hydrostatic approximation of navier-stokes equations

Abstract: Due to the lack of regularity of the solutions to the hydrostatic approximation of Navier–Stokes equations, an energy identity cannot be deduced. By including certain nonlinear perturbations to the hydrostatic approximation equations, the solutions to the perturbed problem are smooth enough so that they satisfy the corresponding energy identity. The perturbations considered in this paper are of the monotone class. Three kinds of problems are then studied. To do that, we introduce a functional setting and show in every case that the set of smooth functions with compact support is dense in the space where we search for solutions. When the perturbations are small enough in a certain sense, the solutions of the perturbed problem are close to those of the original one. As a result, this gives a new proof of the existence of solutions to the hydrostatic approximation of Navier–Stokes equations. Finally, this regularization technique has been applied to the analysis of a one-equation hydrostatic turbulence model.

The Toroidal Iron Atmosphere of a Protoneutron Star: Numerical Solution

Abstract: A numerical method presented by Imshennik et al. (2002) is used to solve the two dimensional axisymmetric hydrodynamic problem on the formation of a toroidal atmosphere during the collapse of an iron stellar core and outer stellar layers. An evolutionary model from Boyes et al. (1999) with a total mass of $25M_{\odot}$ is used as the initial data for the distribution of thermodynamic quantities in the outer shells of a high-mass star. We analyze in detail the results of three calculations in which the difference mesh and the location of the inner boundary of the computational region are varied. In the initial data, we roughly specify an angular velocity distribution that is actually justified by the final result - the formation of a hydrostatic equilibrium toroidal atmosphere with reasonable total mass, $M^{tot} = (0.117 \div 0.122)M_{\odot}$, and total angular momentum, $J^{tot} = (0.445 \div 0.472) x 10^{50} erg \cdot s$, for the two main calculations. We compare the numerical solution with our previous analytical solution in the form of toroidal atmospheres (Imshennik and Manukovskii 2000). This comparison indicates that they are identical if we take into account the more general and complex equation of state with a nonzero temperature and self-gravitation effects in the atmosphere. Our numerical calculations, first, prove the stability of toroidal atmospheres on characteristic hydrodynamic time scales and, second, show the possibility of sporadic fragmentation of these atmospheres even after a hydrodynamic equilibrium is established. The calculations were carried out under the assumption of equatorial symmetry of the problem and up to relatively long time scales $(\approx 10s)$.